Math Problem Statement

tiếng việt

Trong mặt phẳng tọa độ Oxy, cho các đường thẳng d1: x + 2y + 3 = 0, d2: 3x - y + 5 = 0 và điểm P(-2; 1). Đường thẳng Δ đi qua P và cắt d1, d2 lần lượt tại A, B sao cho P là trung điểm của AB. Biết khoảng cách từ M(3; -2) đến đường thẳng Δ có dạng √5. Tính α² + β.

Solution

The uploaded image contains a mathematical problem in Vietnamese. Here's a rough translation:

Question 6: In the coordinate plane OxyOxy, there are two lines:

  • d1:x+2y+3=0d_1: x + 2y + 3 = 0
  • d2:3xy+5=0d_2: 3x - y + 5 = 0 and the point P(2,1)P(-2, 1). The line Δ\Delta passes through PP and intersects both d1d_1 and d2d_2 at points AA and BB, respectively. The point PP is the midpoint of ABAB. The distance from point M(3,2)M(3, -2) to the line Δ\Delta is 5\sqrt{5}. Determine the equation of line Δ\Delta.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Linear Equations
Midpoint Theorem
Distance Formula

Formulas

Equation of a line passing through a point
Midpoint formula
Distance from a point to a line formula

Theorems

Midpoint Theorem
Distance from point to line theorem

Suitable Grade Level

Grades 10-12